4th QUARTER – Module 7: - ZNNHS

Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Zest for Progress

Zeal of Partnership

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4th QUARTER – Module 7: LAW OF COSINES

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

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Mathematics – Grade 9 Alternative Delivery Mode Quarter 4 - Module 7: Law of Cosines First Edition, 2020

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Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio

Printed in the Philippines

Department of Education – Region IX, Zamboanga Peninsula

Office Address: Tiguma, Airport Road, Pagadian City

Telefax: (062) – 215 – 3751; 991 – 5975

E-mail Address: [email protected]

Development Team of the Module

Writer: Shirly V. Gajilomo

Editors: Ma. Pilar C. Ahadi

Mary Rose A. Castillo

Illustrator: Shirly V. Gajilomo

Reviewers: EPS, Mathematics Vilma A. Brown, Ed. D.

Principal Mujim U. Abdurahim

Management Team: SDS Roy C. Tuballa, EMD, JD, CESO VI

ASDS Jay S. Montealto, CESO VI

ASDS Norma T. Francisco, DM, CESE

EPS Mathematics Vilma A. Brown, Ed. D.

EPS LRMS Aida F. Coyme, Ed. D.

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Introductory Message This Self – Learning Module (SLM) is prepared so that you, our dear learners, can continue

your studies and learn while at home. Activities, questions, directions, exercises, and

discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you

discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell

you if you can proceed on completing this module or if you need to ask your facilitator or your

teacher’s assistance for better understanding of the lesson. At the end of each module, you

need to answer the post-test to self-check your learning. Answer keys are provided for each

activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our

facilitators and parents for strategies and reminders on how they can best help you on your

home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use

a separate sheet of paper in answering the exercises and tests. Read the instructions carefully

before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this

module, do not hesitate to consult your teacher or facilitator.

Thank you.

This module will help you understand the concept in solving oblique triangles.

It focuses on law of cosines. As you go through this module, you are expected to

illustrate law of cosines and how it is being applied in solving oblique triangles.

What I Know

Pre-Test: Let us find out how much you know about this module. Choose the letter

that corresponds to your answer. Write your answer on a separate sheet.

For items 1-3, refer to the figure at the right.

1) How long is side d of ∆EDF?

A. 6.13 B. 6.12 C. 6.05 D. 6.01

2) What is m∠F of ∆EDF?

A. 88.300 B. 89.360 C. 91. 450 D. 91.330

What I Need to Know

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3) What is m∠E of ∆EDF?

A. 41.70 B. 40.640 C. 38.550 D. 38.670


4) What is m∠A of ∆ABC?

A. 15.410 B. 16. 230 C. 17.820 D. 18.190

5) Which of the following is the measurement of ∠B?

A. 45.430 B. 51.320 C. 62.220 D. 78.130

LESSON

1 LAW OF COSINES

What’s In

Review Test

Let’s recall the concept on finding the missing parts of a triangle. Choose the letter

that corresponds to your answer. Write your answer on a separate sheet.

1. What is cos 15?

A. 0. 360 B. 0. 480 C. 0. 690 D. 0.970

2. In cos A = 0.2417, what must be the value of A?

A. 76.010 B. 72.090 C. 70.110 D. 68.730


3. What is cos R?

A. -1 B. 0 C. 1 D. 2

4. What is the measure ∠A?

A. 35.670 B. 36.870 C. 40.120 D. 43.410

5. 𝑚∠M = ___.

A. 54.330 B. 53.10 C. 49.880 D. 46.590

In previous lesson, you’ve learned to use trigonometric functions to find the

missing angle or side in a right triangle.

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What’s New

Problem:

John constructed a triangular table to be placed in the Math Park. The said

table has sides of 147cm and 166cm and its included angle is 500. How long is the

remaining side of the table? (Round off to the nearest whole number)

In this lesson, you will learn to find the missing side or angle of an oblique

triangle using law of cosines.

What is It

Oblique triangles can also be solved using the law of cosines.

Law of cosines c2 = a2 + b2 – 2ab (cosC) is an extension of Pythagorean

theorem because if 𝜃 is a right angle, then it becomes c2 = a2 + b2

Law of Cosines is applied when the following information are given:

Case 1: two sides and an included angle or Side-Angle-Side (SAS) Case 2: three sides or Side-Side-Side (SSS)

In any ΔABC, the square of the length of one side is equal to the sum of the squares of the other two sides minus the product of twice the two sides and the cosine of the angle between them. In symbols, a2 = b2 + c2 – 2bc (cosA), b2 = a2 + c2 – 2ac (cosB), c2 = a2 + b2 – 2ab (cosC)

LAW of COSINES

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Determine the measure of the missing parts of ΔDAY.

Solution: By Law of Cosines A. Solve for side y

Steps Illustrations

1) Identify the formula to solve for side y based on law of cosines.

y2 = d2 + a2 – 2da (cosY)

2) Substitute the given values y2 = (2)2 + (3)2 – 2(2)(3) (cos640)

3) Simplify the equation

y2 = 4 + 9 – 12 (0.4384) y2 = 4 + 9 – 5.2608 y2 = 13 – 5.2608 y2 = 7.7392

4) Extract the square root and simplify the result

ඥ𝑦2 = ξ7.7392 → y = 2.78 cm

B. Solve for ∠𝑨

Steps Illustrations

1) Identify the formula to solve for ∠𝑨 based on law of cosines.

a2 = y2 + d2 – 2yd (cosA)

2) Substitute the given values (3)2 = (2.78)2 + (2)2 – 2(2.78)(2) (cosA)


9 = 7.7284 + 4 – 11.12 (cosA) 9 = 11.7284 – 11.12 (cosA) 11.12 (cosA) = 11.7284 – 9 11.12 (𝑐𝑜𝑠𝐴)

11.12=

2.7284

11.12

cos A = 0.2454

4) Multiply the inverse of cosine (arccos or cos-1) to both sides of the equation

cos-1 (cos A) = cos-1 (0.2454)

5) Simplify the resulting equation A = 75.79 m∠𝑨 = 75.790

C. Solve for ∠D

Steps Illustrations

1) Identify the formula to solve for ∠𝐷. Apply the concept “the sum of the three interior angles in a triangle is always 180°.”

𝑚∠𝐷 + 𝑚∠𝐴 + 𝑚∠𝑌 = 1800

2) Substitute the given values 𝑚∠𝐷 + 75.790 + 640 = 1800

3) Simplify the left side of the equation 𝑚∠𝐷 + 139.790 = 1800

4) Apply Addition Property of Equality 𝑚∠𝐷 +139.790+(-139.790) = 1800+(-139.790)

5) Simplify the value of ∠𝐷 𝑚∠𝐷 = 40.210

ILLUSTRATIVE EXAMPLE 1: CASE 1

Given: Two sides and the included angle

𝑚∠𝑌 = 640 , 𝑠𝑖𝑑𝑒 𝑑 = 2𝑐𝑚 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 𝑎 = 3 𝑐𝑚.

Solve the measurement of: side y, ∠𝐴 and ∠D

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Determine the measure of the missing parts of ΔFLY.

Solution: By Law of Cosines

A. Solve for ∠𝑭

Steps Illustrations

1) Identify the formula to solve for ∠𝐹 based on law of cosines.

f2 = y2 + l2 – 2yl (cosF)

2) Substitute the given values (7)2 = (10)2 + (5)2 – 2(10)(5) (cosF)


49 = 100 + 25 – 100 (cosF) 49 = 125 – 100 (cosF) 100 (cosF) = 125 – 49 100 (𝑐𝑜𝑠𝐹)

100=

76

100

cos F = 0.76


cos-1 (cos F) = cos-1 (0.76)

5) Simplify the resulting equation F = 40.54 m∠𝐹 = 40.540

B. Solve for ∠𝑳

Steps Illustrations

1) Identify the formula to solve for ∠𝑳 based on law of cosines.

l2 = f2 + y2 – 2fy (cosL)

2) Substitute the given values (5)2 = (7)2 + (10)2 – 2(7)(10) (cosL)


25 = 49 + 100 – 140 (cosL) 25 = 149 – 140 (cosL) 140 (cosL)= 149 – 25 140 (cosL)

140=

124

140

cos L = 0.8857


cos-1 (cos L) = cos-1 (0.8857)

5) Simplify the resulting equation L = 27.66 m∠𝐿 = 27.660

C. Solve for ∠Y

Steps Illustrations

1) Identify the formula to solve for ∠𝑌. Apply the concept “the sum of the three interior angles in a triangle is always 180°.”

𝑚∠𝐹 + 𝑚∠𝐿 + 𝑚∠𝑌 = 1800

2) Substitute the given values 40.540 + 27.660+ 𝑚∠𝑌 = 1800

3) Simplify the left side of the equation 𝑚∠𝑌 + 68.20 = 1800

4) Apply Addition Property of Equality 𝑚∠𝑌 + 68.20+(-68.20) = 1800+(-68.20)

5) Simplify the value of ∠𝑌 𝑚∠𝑌 = 111.80

ILLUSTRATIVE EXAMPLE 2: CASE 2

Given: Three sides 𝑠𝑖𝑑𝑒 𝑦 = 10𝑐𝑚, 𝑠𝑖𝑑𝑒 𝑓 = 7𝑐𝑚 𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 𝑙 = 5 𝑐𝑚.

Solve the measurement of: ∠𝐹, ∠𝐿 and ∠Y

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What’s More

Directions: Solve for the missing measurement of side/angle in each triangle. Identify the corresponding item number on the answer box below to reveal the pincode. Write your answer on a separate sheet.

1) side c = _______ cm

3) 𝑚∠H = _______ 0

2) side d = _______ cm

4) 𝑚∠J = _____ 0

Pin code

Answers 33.56 9.70 88.97 5.21

What I Have Learned

Directions: Fill in the blank/s to make the statement true. Write your answer on a

separate sheet.

1. A law which states that the square of the length of one side is equal to the sum of

the squares of the other two sides minus the product of twice the two sides and the

cosine of the angle between them is called ___.

2. In ΔABC, law of cosines can be expressed in symbols as _______.

3. Law of cosines is applied when ___ or ___ are known in a triangle.

4. The ___ of the interior angles of a triangle is 1800.

5. If cos 𝐴 = 0, then 𝐴 must be a/an ____ angle.

What’s the Pin code? ACTIVITY

Fill Me! ACTIVITY

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What I Can Do

Directions: Solve the problem. Write your answer on a separate sheet.

Assessment

Directions: Read and understand each item carefully. Choose the letter that

corresponds to your answer. Write your answer on a separate sheet.

1. Which of the following equations describes law of cosines?

A. sin 𝐴

𝑎=

sin 𝐵

𝑏=

sin 𝐶

𝑐 B. c2 = a2 + b2

C. a2 = b2 + c2 – 2bc (cosA) D. y = ax2 + bx + c

2. Which of the following conditions is applicable for law of cosines?

I. SSS II. SAS III. ASA IV. SAA V. SSA

A. I and II B. III, IV and V C. II and V D. III and IV

For items 3-6, refer to △ 𝑫𝑬𝑭 at the right.

3. What is the length of side e?

A. 6. 72 B. 6. 25 C. 5. 84 D. 5.75

4. What is m∠D?

A. 62.05 B. 61.07 C. 62.24 D. 63.34

5. What is m∠F?

A. 78.95 B. 79.93 C. 78.76 D. 77.66

6. Which angle has the largest measurement?

A. ∠D B. ∠E C. ∠F D. cannot be determined

For items 7-10, refer to △ 𝑳𝑴𝑵 at the right.

7. Which of the following is the measurement of ∠L?

A. 83.33 B. 67.31 C. 52.62 D. 44.04

8. What is the measure of ∠M?

A. 83.33 B. 67.31 C. 52.62 D. 44.04

9. 𝑚∠N = ___

A. 83.33 B. 67.31 C. 52.62 D. 44.04

10. Which angle has the smallest measurement?

A. ∠L B. ∠M C. ∠N D. cannot be determined

Solve me! ACTIVITY

Based on the data shown at the right, Jose wants to add more clothesline in his backyard. How many meters of rope are needed to connect from post A to post B? (Round your answer to the nearest whole number)

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Answer Key

What I Know:

1. A 2. D 3. D 4. D 5. B

What’s In:

1. D 2. A 3. B 4. B 5. B

What’s New:

133cm

What’s More:

Pin code: 3 1 4 2

What I Have Learned:

1. Law of cosines 2. a

2 = b

2 + c

2 – 2bc (cosA) or b

2 = a

2 + c

2 – 2ac (cosB) or

c2 = a

2 + b

2 – 2ab (cosC)

3. SAS, SSS 4. sum 5. right

b2 = a2 + c2 – 2ac (cosB),

What I Can Do:

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Assessment:

1. C 2. A 3. D 4. B 5. B 6. C 7. C 8. A 9. D 10. C

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Reference:

Bryant, Merden, Mathematics 9 Learner’s Material. Pasig City: Department of Education,

2014.

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I AM A FILIPINO

by Carlos P. Romulo

I am a Filipino – inheritor of a glorious past, hostage to the

uncertain future. As such, I must prove equal to a two-fold

task – the task of meeting my responsibility to the past, and

the task of performing my obligation to the future.

I am sprung from a hardy race – child many generations

removed of ancient Malayan pioneers. Across the centuries,

the memory comes rushing back to me: of brown-skinned

men putting out to sea in ships that were as frail as their hearts

were stout. Over the sea I see them come, borne upon the

billowing wave and the whistling wind, carried upon the

mighty swell of hope – hope in the free abundance of the new

land that was to be their home and their children’s forever.

This is the land they sought and found. Every inch of shore

that their eyes first set upon, every hill and mountain that

beckoned to them with a green and purple invitation, every

mile of rolling plain that their view encompassed, every river

and lake that promised a plentiful living and the fruitfulness

of commerce, is a hollowed spot to me.

By the strength of their hearts and hands, by every right of

law, human and divine, this land and all the appurtenances

thereof – the black and fertile soil, the seas and lakes and

rivers teeming with fish, the forests with their inexhaustible

wealth in wild and timber, the mountains with their bowels

swollen with minerals – the whole of this rich and happy land

has been for centuries without number, the land of my

fathers. This land I received in trust from them, and in trust

will pass it to my children, and so on until the world is no

more.

I am a Filipino. In my blood runs the immortal seed of heroes

– seed that flowered down the centuries in deeds of courage

and defiance. In my veins yet pulses the same hot blood that

sent Lapulapu to battle against the alien foe, that drove Diego

Silang and Dagohoy into rebellion against the foreign

oppressor.

That seed is immortal. It is the self-same seed that flowered

in the heart of Jose Rizal that morning in Bagumbayan when

a volley of shots put an end to all that was mortal of him and

made his spirit deathless forever; the same that flowered in

the hearts of Bonifacio in Balintawak, of Gregorio del Pilar

at Tirad Pass, of Antonio Luna at Calumpit, that bloomed in

flowers of frustration in the sad heart of Emilio Aguinaldo at

Palanan, and yet burst forth royally again in the proud heart

of Manuel L. Quezon when he stood at last on the threshold

of ancient Malacanang Palace, in the symbolic act of

possession and racial vindication. The seed I bear within me

is an immortal seed.

It is the mark of my manhood, the symbol of my dignity as

a human being. Like the seeds that were once buried in the

tomb of Tutankhamen many thousands of years ago, it shall

grow and flower and bear fruit again. It is the insigne of my

race, and my generation is but a stage in the unending

search of my people for freedom and happiness.

I am a Filipino, child of the marriage of the East and the

West. The East, with its languor and mysticism, its passivity

and endurance, was my mother, and my sire was the West

that came thundering across the seas with the Cross and

Sword and the Machine. I am of the East, an eager

participant in its struggles for liberation from the imperialist

yoke. But I know also that the East must awake from its

centuried sleep, shake off the lethargy that has bound its

limbs, and start moving where destiny awaits.

For I, too, am of the West, and the vigorous peoples of the

West have destroyed forever the peace and quiet that once

were ours. I can no longer live, a being apart from those

whose world now trembles to the roar of bomb and cannon

shot. For no man and no nation is an island, but a part of the

main, and there is no longer any East and West – only

individuals and nations making those momentous choices

that are the hinges upon which history revolves. At the

vanguard of progress in this part of the world I stand – a

forlorn figure in the eyes of some, but not one defeated and

lost. For through the thick, interlacing branches of habit and

custom above me I have seen the light of the sun, and I

know that it is good. I have seen the light of justice and

equality and freedom, my heart has been lifted by the vision

of democracy, and I shall not rest until my land and my

people shall have been blessed by these, beyond the power

of any man or nation to subvert or destroy.

I am a Filipino, and this is my inheritance. What pledge

shall I give that I may prove worthy of my inheritance? I

shall give the pledge that has come ringing down the

corridors of the centuries, and it shall be compounded of the

joyous cries of my Malayan forebears when first they saw

the contours of this land loom before their eyes, of the battle

cries that have resounded in every field of combat from

Mactan to Tirad Pass, of the voices of my people when they

sing:

“I am a Filipino born to freedom, and I shall not rest until

freedom shall have been added unto my inheritance—for

myself and my children and my children’s children—

forever.”

4th QUARTER – Module 7: - ZNNHS

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